Quantifying the checkerboard problem to reduce numerical dissipation

This work provides a comprehensive exploration of various methods in solving incompressible flows using a projection method, and their relation to the occurrence and management of checkerboard oscillations. It employs an algebraic symmetry-preserving framework, clarifying the derivation and implementation of discrete operators while also addressing the associated numerical errors. The lack of a proper definition for the checkerboard problem is addressed by proposing a physics-based coefficient. This coefficient, rooted in the disparity between the compact- and wide-stencil Laplacian operators, is able to quantify oscillatory solution fields with a physics-based, global, normalised, non-dimensional value. The influence of mesh and time-step refinement on the occurrence of checkerboarding is highlighted. Therefore, single measurements using this coefficient should be considered with caution, as the value presents little use without any context and can either suggest mesh refinement or use of a different solver. In addition, an example is given on how to employ this coefficient, by establishing a negative feedback between the level of checkerboarding and the inclusion of a pressure predictor, to dynamically balance the checkerboarding and numerical dissipation. This method is tested for laminar and turbulent flows, demonstrating its capabilities in obtaining this dynamical balance, without requiring user input. The method is able to achieve low numerical dissipation in absence of oscillations or diminish oscillation on skew meshes, while it shows minimal loss in accuracy for a turbulent test case. Despite its advantages, the method exhibits a slight decrease in the second-order relation between time-step size and pressure error, suggesting that other feedback mechanisms could be of interest.